nhaliday + heavyweights + πŸŽ“   15

soft question - Thinking and Explaining - MathOverflow
- good question from Bill Thurston
- great answers by Terry Tao, fedja, Minhyong Kim, gowers, etc.

Terry Tao:
- symmetry as blurring/vibrating/wobbling, scale invariance
- anthropomorphization, adversarial perspective for estimates/inequalities/quantifiers, spending/economy

fedja walks through his though-process from another answer

Minhyong Kim: anthropology of mathematical philosophizing

Per Vognsen: normality as isotropy
comment: conjugate subgroup gHg^-1 ~ "H but somewhere else in G"

gowers: hidden things in basic mathematics/arithmetic
comment by Ryan Budney: x sin(x) via x -> (x, sin(x)), (x, y) -> xy
I kinda get what he's talking about but needed to use Mathematica to get the initial visualization down.
To remind myself later:
- xy can be easily visualized by juxtaposing the two parabolae x^2 and -x^2 diagonally
- x sin(x) can be visualized along that surface by moving your finger along the line (x, 0) but adding some oscillations in y direction according to sin(x)
q-n-a  soft-question  big-list  intuition  communication  teaching  math  thinking  writing  thurston  lens  overflow  synthesis  hi-order-bits  πŸ‘³  insight  meta:math  clarity  nibble  giants  cartoons  gowers  mathtariat  better-explained  stories  the-trenches  problem-solving  homogeneity  symmetry  fedja  examples  philosophy  big-picture  vague  isotropy  reflection  spatial  ground-up  visual-understanding  polynomials  dimensionality  math.GR  worrydream  scholar  πŸŽ“  neurons  metabuch  yoga  retrofit  mental-math  metameta  wisdom  wordlessness  oscillation  operational  adversarial  quantifiers-sums  exposition  explanation  tricki  concrete  s:***  manifolds  invariance  dynamical  info-dynamics  cool  direction  elegance  heavyweights  analysis  guessing  grokkability-clarity  technical-writing 
january 2017 by nhaliday
soft question - How do you not forget old math? - MathOverflow
Terry Tao:
I find that blogging about material that I would otherwise forget eventually is extremely valuable in this regard. (I end up consulting my own blog posts on a regular basis.) EDIT: and now I remember I already wrote on this topic: terrytao.wordpress.com/career-advice/write-down-what-youve-dβ€Œβ€‹one

fedja:
The only way to cope with this loss of memory I know is to do some reading on systematic basis. Of course, if you read one paper in algebraic geometry (or whatever else) a month (or even two months), you may not remember the exact content of all of them by the end of the year but, since all mathematicians in one field use pretty much the same tricks and draw from pretty much the same general knowledge, you'll keep the core things in your memory no matter what you read (provided it is not patented junk, of course) and this is about as much as you can hope for.

Relating abstract things to "real life stuff" (and vice versa) is automatic when you work as a mathematician. For me, the proof of the Chacon-Ornstein ergodic theorem is just a sandpile moving over a pit with the sand falling down after every shift. I often tell my students that every individual term in the sequence doesn't matter at all for the limit but somehow together they determine it like no individual human is of any real importance while together they keep this civilization running, etc. No special effort is needed here and, moreover, if the analogy is not natural but contrived, it'll not be helpful or memorable. The standard mnemonic techniques are pretty useless in math. IMHO (the famous "foil" rule for the multiplication of sums of two terms is inferior to the natural "pair each term in the first sum with each term in the second sum" and to the picture of a rectangle tiled with smaller rectangles, though, of course, the foil rule sounds way more sexy).

One thing that I don't think the other respondents have emphasized enough is that you should work on prioritizing what you choose to study and remember.

Timothy Chow:
As others have said, forgetting lots of stuff is inevitable. But there are ways you can mitigate the damage of this information loss. I find that a useful technique is to try to organize your knowledge hierarchically. Start by coming up with a big picture, and make sure you understand and remember that picture thoroughly. Then drill down to the next level of detail, and work on remembering that. For example, if I were trying to remember everything in a particular book, I might start by memorizing the table of contents, and then I'd work on remembering the theorem statements, and then finally the proofs. (Don't take this illustration too literally; it's better to come up with your own conceptual hierarchy than to slavishly follow the formal hierarchy of a published text. But I do think that a hierarchical approach is valuable.)

Organizing your knowledge like this helps you prioritize. You can then consciously decide that certain large swaths of knowledge are not worth your time at the moment, and just keep a "stub" in memory to remind you that that body of knowledge exists, should you ever need to dive into it. In areas of higher priority, you can plunge more deeply. By making sure you thoroughly internalize the top levels of the hierarchy, you reduce the risk of losing sight of entire areas of important knowledge. Generally it's less catastrophic to forget the details than to forget about a whole region of the big picture, because you can often revisit the details as long as you know what details you need to dig up. (This is fortunate since the details are the most memory-intensive.)

Having a hierarchy also helps you accrue new knowledge. Often when you encounter something new, you can relate it to something you already know, and file it in the same branch of your mental tree.
thinking  math  growth  advice  expert  q-n-a  πŸŽ“  long-term  tradeoffs  scholar  overflow  soft-question  gowers  mathtariat  ground-up  hi-order-bits  intuition  synthesis  visual-understanding  decision-making  scholar-pack  cartoons  lens  big-picture  ergodic  nibble  zooming  trees  fedja  reflection  retention  meta:research  wisdom  skeleton  practice  prioritizing  concrete  s:***  info-dynamics  knowledge  studying  the-trenches  chart  expert-experience  quixotic  elegance  heavyweights 
june 2016 by nhaliday
Answer to What is it like to understand advanced mathematics? - Quora
thinking like a mathematician

some of the points:
- small # of tricks (echoes Rota)
- web of concepts and modularization (zooming out) allow quick reasoning
- comfort w/ ambiguity and lack of understanding, study high-dimensional objects via projections
- above is essential for research (and often what distinguishes research mathematicians from people who were good at math, or majored in math)
math  reflection  thinking  intuition  expert  synthesis  wormholes  insight  q-n-a  πŸŽ“  metabuch  tricks  scholar  problem-solving  aphorism  instinct  heuristic  lens  qra  soft-question  curiosity  meta:math  ground-up  cartoons  analytical-holistic  lifts-projections  hi-order-bits  scholar-pack  nibble  the-trenches  innovation  novelty  zooming  tricki  virtu  humility  metameta  wisdom  abstraction  skeleton  s:***  knowledge  expert-experience  elegance  judgement  advanced  heavyweights  guessing 
may 2016 by nhaliday
Work hard | What's new
Similarly, to be a β€œprofessional” mathematician, you need to not only work on your research problem(s), but you should also constantly be working on learning new proofs and techniques, going over important proofs and papers time and again until you’ve mastered them. Don’t stay in your mathematical comfort zone, but expand your horizon by also reading (relevant) papers that are not at the heart of your own field. You should go to seminars to stay current and to challenge yourself to understand math in real time. And so on. All of these elements have to find their way into your daily work routine, because if you neglect any of them it will ultimately affect your research output negatively.
- from the comments
advice  academia  math  reflection  career  expert  gowers  long-term  πŸŽ“  aphorism  grad-school  phd  scholar  mathtariat  discipline  curiosity  πŸ¦‰  nibble  org:bleg  the-trenches  meta:research  gtd  stamina  vitality  s:**  info-dynamics  expert-experience  heavyweights 
april 2016 by nhaliday

bundles : academe ‧ growth ‧ stars

related tags

abstraction βŠ•  academia βŠ•  acmtariat βŠ•  advanced βŠ•  adversarial βŠ•  advice βŠ•  aesthetics βŠ•  analogy βŠ•  analysis βŠ•  analytical-holistic βŠ•  aphorism βŠ•  applications βŠ•  berkeley βŠ•  better-explained βŠ•  big-list βŠ•  big-picture βŠ•  bret-victor βŠ•  career βŠ•  cartoons βŠ•  chart βŠ•  checklists βŠ•  clarity βŠ•  communication βŠ•  compilers βŠ•  computation βŠ•  concrete βŠ•  contrarianism βŠ•  cool βŠ•  coordination βŠ•  cs βŠ•  culture βŠ•  curiosity βŠ•  debt βŠ•  decision-making βŠ•  dimensionality βŠ•  direction βŠ•  discipline βŠ•  discussion βŠ•  dynamical βŠ•  education βŠ•  elegance βŠ•  ergodic βŠ•  essay βŠ•  examples βŠ•  expert βŠ•  expert-experience βŠ•  explanation βŠ•  exposition βŠ•  fedja βŠ•  formal-methods βŠ•  frontier βŠ•  geometry βŠ•  giants βŠ•  gowers βŠ•  grad-school βŠ•  grokkability-clarity βŠ•  ground-up βŠ•  growth βŠ•  gtd βŠ•  guessing βŠ•  heavyweights βŠ–  heuristic βŠ•  hi-order-bits βŠ•  homogeneity βŠ•  humility βŠ•  ideas βŠ•  impact βŠ•  inference βŠ•  info-dynamics βŠ•  info-foraging βŠ•  innovation βŠ•  insight βŠ•  instinct βŠ•  interdisciplinary βŠ•  intuition βŠ•  invariance βŠ•  isotropy βŠ•  judgement βŠ•  knowledge βŠ•  learning βŠ•  lens βŠ•  lifts-projections βŠ•  links βŠ•  list βŠ•  local-global βŠ•  long-term βŠ•  manifolds βŠ•  math βŠ•  math.GR βŠ•  mathtariat βŠ•  mental-math βŠ•  meta:math βŠ•  meta:reading βŠ•  meta:research βŠ•  meta:science βŠ•  metabuch βŠ•  metameta βŠ•  michael-nielsen βŠ•  multi βŠ•  narrative βŠ•  neurons βŠ•  nibble βŠ•  novelty βŠ•  operational βŠ•  org:bleg βŠ•  org:edu βŠ•  org:popup βŠ•  oscillation βŠ•  overflow βŠ•  p:whenever βŠ•  papadimitriou βŠ•  papers βŠ•  pdf βŠ•  phd βŠ•  philosophy βŠ•  polynomials βŠ•  practice βŠ•  prioritizing βŠ•  problem-solving βŠ•  productivity βŠ•  programming βŠ•  proofs βŠ•  q-n-a βŠ•  qra βŠ•  quantifiers-sums βŠ•  quixotic βŠ•  reflection βŠ•  research βŠ•  retention βŠ•  retrofit βŠ•  rhetoric βŠ•  rigor βŠ•  s-factor βŠ•  s:* βŠ•  s:** βŠ•  s:*** βŠ•  scholar βŠ•  scholar-pack βŠ•  science βŠ•  skeleton βŠ•  social βŠ•  soft-question βŠ•  spatial βŠ•  stamina βŠ•  stories βŠ•  strategy βŠ•  studying βŠ•  success βŠ•  survey βŠ•  symmetry βŠ•  synthesis βŠ•  tactics βŠ•  tcs βŠ•  teaching βŠ•  tech βŠ•  technical-writing βŠ•  techtariat βŠ•  the-trenches βŠ•  thinking βŠ•  thurston βŠ•  tradeoffs βŠ•  trees βŠ•  tricki βŠ•  tricks βŠ•  trust βŠ•  unit βŠ•  vague βŠ•  vazirani βŠ•  virtu βŠ•  visual-understanding βŠ•  vitality βŠ•  wisdom βŠ•  wordlessness βŠ•  wormholes βŠ•  worrydream βŠ•  writing βŠ•  yoga βŠ•  zooming βŠ•  πŸŽ“ βŠ–  πŸ‘³ βŠ•  πŸ”¬ βŠ•  πŸ–₯ βŠ•  πŸ¦‰ βŠ• 

Copy this bookmark:



description:


tags: